162 BELL SYSTEM TECHNICAL JOURNAL 



II. Solution of Maxwell's Equations In Tensor Form 



In tensor notation, Maxwell's equations for a nonmagnetic medium with 

 no free charges take the form 



1 dDi _ dHj 1 dHj _ dEk dDi _ . dHj _ . ,^. 



V dt dx'k V dt dXi dXi dXj 



where Dt is the electric displacement, H_, the magnetic field, Eu the electric 

 field, V the velocity of light in vacuo and ^ijk a tensor equal to zero when 

 i = j or k ox j — k, but equal to 1 or — 1 when all three numbers are different. 

 If the numbers are in rotation, i.e. 1, 2, 3; 2, 3, 1; 3, 1, 2 the value is +1 

 while, if they are out of rotation, the value is —1. 



We assume the electric vector to be representable by a plane wave whose 

 planes of equal phase are taken normal to the unit vector »j . Then 



£, = Eo.e^"^'""'"'"'^ (2) 



where Eo^ are constants representing the maximum values of the field along 

 the three rectangular coordinates and 7 — v — 1. Substituting (2) in the 

 second of equations (1), noting that Eof. are not functions of the space co- 

 ordinates, we have 



1 dHj Jiii r -, ji^U-XiUilv] ,2\ 



Tr^T = -~ [ejkiEo^mle . (3) 



V dt V 



Integrating with respect to the time 



Hj — — [ejiciEoi^nile = Ha-e . (.4; 



Hence, 



^oy = - ka-£oi«i] (5) 



V 



and therefore the magnetic vector is normal to the plane determined by 

 £oi and Ui . 



Next, using the first of equations (1), 



dt dXk dXk 



(6) 



— [(ijh lit), iit\e 



V ' 



Integrating with respect to time, 



